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Topic: Algebra over a field

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	Algebra over a field (Site not responding. Last check: 2007-11-07)
		In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A.
		Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K-algebra, and K is the base ring of A.
		For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A.
	www.sciencedaily.com /encyclopedia/algebra_over_a_field (1149 words)

	Algebra - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		Algebra is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic; it also refers to a particular kind of abstract algebra structure, the algebra over a field.
		* abstract algebra, where algebraic structures such as groups, rings and as fields are axiomatically defined and investigated.
		In advanced studies axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural topology compatible with algebraic structure.
	encyclopedia.worldsearch.com /algebra.htm (806 words)

	Algebra over a field Article, Algebraoverfield Information (Site not responding. Last check: 2007-11-07)
		In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion ofmultiplication of elements of A.
		Algebras can also more generally be defined over any commutativering K: we need a module A overK and a bilinear multiplication operation which satisfies the same identities as above; then A is aK-algebra, and K is the base ring of A.
		A commutative algebra is one whose multiplication is commutative ; an associative algebra is onewhose multiplication is associative.
	www.anoca.org /algebras/multiplication/algebra_over_a_field.html (1064 words)

	[No title]
		The matrix algebra M(n,R) is an algebra over the field R. An algebraic number field +------------------------------------------------------------ An algebraic number field is a subfield of the complex numbers that arises as a finite degree algebraic extension field over the field of rationals.
		It is a field extension of Q which is also a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of a fixed polyonomial a_0+a_1 z+...
		It is in general a field extension of degree 2 over the field of rational number.
	www.math.harvard.edu /~knill/sofia/data/algebra.txt (1599 words)

	New Simple Lie algebras: Melikyan Algebras (Site not responding. Last check: 2007-11-07)
		Over the field of complex numbers, the simple Lie algebras were found by Killing-Cartan.
		The theory of finite-dimensional Lie algebras over fields of positive characteristic p was initiated by E. Witt, N. Jacobson [5].
		Kostrikin and A. umadildaev [14] is the construction of Melikyan algebras as deformations of certain Hamiltonian Lie algebras.
	www.nccu.edu /artsci/math/melikyan/res/node2.html (1082 words)

	Algebra biography .ms (Site not responding. Last check: 2007-11-07)
		Algebra (from the Arabic "al-jabr" meaning "reunion", "connection" or "completion") is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic; it also refers to a particular kind of abstract algebra structure, the algebra over a field.
		elementary algebra, where the properties of operations on the real number system are recorded, symbols are used as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied,
		abstract algebra, where algebraic structures such as groups, rings and as fields are axiomatically defined and investigated.
	algebra.biography.ms (733 words)

	Associative algebra - FreeEncyclopedia (Site not responding. Last check: 2007-11-07)
		An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A
		The dimension of the associative algebra A over the field K is its dimension as a K-vector space.
		Both are algebras over R, and the map which assigns to every continuous function f the number f(0) is an algebra homomorphism from A to B.
	openproxy.ath.cx /as/Associative_algebra.html (612 words)

	Division algebra (Site not responding. Last check: 2007-11-07)
		In abstract algebra, a division algebra is a unitary associative algebra with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e.
		The prototypical example of a division algebra over the real numbers is given by the quaternions.
		Whenever A is an associative algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every division algebra over F arises in this fashion.
	www.city-search.org /di/division-algebra.html (493 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-11-07)
		Similarly, for a "p-restricted Lie algebra" g over a field k of characteristic p>0, the restricted representations of g are the modules for the restricted enveloping algebra V(g).
		For \pi a finite group and g a finite dimensional p-restricted Lie algebra, the algebras k\pi and V(g) are examples of a general class of algebras, the group algebras of finite group schemes.
		One means of study of representations is to use cohomology which introduces algebro-geometric invariants associated to the commutative k algebra of even dimensional cohomology.
	www.math.unl.edu /pi/colloquia/abstract-20021003.txt (181 words)

	Introduction (Site not responding. Last check: 2007-11-07)
		A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one.
		In the literature such an algebra is known as a "split" basic algebra.
		The algebra A is generated by elements a_1, a_2,..., a_t where a_1,..., a_s are the primitive idempotent generators and a_(s + 1),..., a_t are the nonidempotent generators.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1007.htm (129 words)

	Vector (spatial) - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry).
		Vectors are the building blocks of vector fields and vector calculus.
		Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations).
	www.sterlingheights.us /project/wikipedia/index.php/Vector_(spatial) (2016 words)

	Creation of Quaternion Algebras
		A general constructor for a quaternion algebra over any field K creates a model in terms of two generators x and y and three relations x^2 = a, y^2 = b, and xy = - yx.
		This function creates the quaternion algebra A over the field K on generators i and j with relations i^2 = a, j^2 = b, and ij = - ji.
		The fact that the latter Kronecker symbols are -1, indicating that 41 is inert in the quadratic fields of discriminants -7, -47, and -95, proves that 41 is a ramified prime, and 2 is not.
	www.math.lsu.edu /magma/text865.htm (683 words)

	Gilles Villard (Site not responding. Last check: 2007-11-07)
		One may consider that the algebraic complexity of basic linear algebra over an abstract field K is well known.
		Indeed, if w is the exponent of matrix multiplication over a field K, for instance computing the determinant, the matrix inverse, the rank or the characteristic polynomial of an n x n matrix can be done in softO(n^w) operations in K (the same holds for the solution of many other problems).
		For reducing the complexity, a main concern is to exploit the interplay of the algebraic structure with the intermediate expression swell.
	www.siam.org /meetings/la03/invited/villard.htm (301 words)

	Articles - Algebra over a field (Site not responding. Last check: 2007-11-07)
		Two algebras A and B over K are isomorphic if there exists a bijective K-linear map f : A â’ B such that f(xy) = f(x) f(y) for all x,y in A.
		When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations.
		algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane.
	www.lastring.com /articles/Algebra_over_a_field?mySession=ada356ad5465ee7cbd50c09526c0abdb (1324 words)

	Introduction (Site not responding. Last check: 2007-11-07)
		An affine algebra in Magma is simply the quotient ring of a multivariate polynomial ring P = R[x_1,..., x_n] by an ideal J of P. Such rings arise commonly in commutative algebra and algebraic geometry.
		The elements of affine algebras are simply multivariate polynomials which are always kept reduced to normal form modulo the ideal J of "relations".
		If an affine algebra defined over a field has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.
	www.math.lsu.edu /magma/text1134.htm (209 words)

	Dear Stefan:
		These Hopf algebras are also known as the restricted specializations of quantized enveloping algebras associated to the semisimple Lie algebra algebra g.
		In fact this Hopf algebra is the degree zero part of a larger one which is based on all the planar trees.
		Suppose that $H$ is a finite-dimensional Hopf algebra over a field $k$.
	condor.depaul.edu /~scatoiu/seminar/february2002/abstracts.html (1659 words)

	Lakhdar Hammoudi, January 24, 2003 (Site not responding. Last check: 2007-11-07)
		This question gave birth to the study of finiteness conditions in algebra and led to the proliferation of Burnside type problems.
		Recently Smoktunowicz constructed a 3 generated nil algebra over a denumerable field such that its algebra of polynomials with two commuting indeterminates is not nil.
		Our algebras are also different from Smoktunowicz's as we carry out our construction over every field, which leads to absolutely nil algebras.
	www.math.ohiou.edu /~slopez/lakhdar.html (330 words)

	[No title]
		Note that when is a (graded) cocommutative Hopf algebra over a field k, then H(; k) is a (graded) commutative k-algebra_one can use the diagonal map on to induce the product on H(; k), so cocommutativity of implies commutativity of H*(; k).
		Example 3.3.Suppose that is a cocommutative Hopf algebra over a field k, and let * be the dual of, with the standard left -module structure (i.e., if k is * the only simple -module, then is the injective hull of k).
		Proposition 3.4.Fix a graded connected cocommutative Hopf algebra over a field k, and fix a property P which is generic on -modules.
	hopf.math.purdue.edu /Palmieri/palmieri-f-iso.txt (3541 words)

	[tut] 6.2 Algebras
		Of course the coefficient field and the generators must fit together; if we want to construct an algebra of ordinary matrices, we may take the field generated by the entries of the generating matrices, or a subfield or extension field.
		The second number that is added is the zero element of the field over which the algebra is defined.
		If the coefficient field is a real subfield of the complex numbers then the quaternion algebra is in fact a division ring.
	www.mathematik.uni-kassel.de /gap4/tut/C006S002.htm (1314 words)

	Affine Algebras which are Fields (Site not responding. Last check: 2007-11-07)
		If the ideal J of relations defining an affine algebra A = K[x_1,..., x_n]/J, where K is a field, is maximal, then A is a field and may be used with any algorithms in Magma which work over fields.
		Note that an affine algebra defined over a field which itself is a field also has finite dimension when considered as a vector space over its coefficient field, so all of the operations in the previous section are also available.
		Starting with the same affine algebra A = Q(a, b, x)F[y]/ as in the last example, we factor some univariate polynomials over A. A is of course isomorphic to an absolute field, but the presentation given may be much more convenient to the user.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1139.htm (441 words)

	Vector space (Site not responding. Last check: 2007-11-07)
		A vector space over the field of complex numbers C''' is called a '''complex vector space.
		Given a translationally invariant and rescaling invariant topology over a vector space (preferably infinite-dimensional), the sum of an infinite sequence of vectors can be defined as the limit (topology)topological limit, if it exists.
		Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products.
	www.infothis.com /find/Vector_space (1274 words)

	QuickMath Automatic Math Solutions
		The tem 'algebra' is used for many things in mathematics, but in this section we'll just be talking about the sort of algebra you come across at high-school.
		Algebra is the branch of elementary mathematics which uses symbols to stand for unknown quantities.
		Although solving equations is really a part of algebra, it is such a big area that it has its own section in QuickMath.
	www.quickmath.com /www02/pages/modules/algebra/index.shtml (642 words)

	Dear Stefan:
		Abstract: The gradings of associative algebras by groups has proven to be an extremely useful tool in examining the invariants of derivations and automorphisms.
		Many of the results for gradings of associative algebras do not hold in this situation and we discuss some results on gradings which do hold in this situation and can be used to study the invariants of derivations and automorphisms.
		Abstract: Table algebras are finite dimensional algebras over the complex numbers with a distinguished basis whose properties generalize those of group algebras, centers of group algebras, double coset algebras, character rings, and the adjacency algebras derived from association schemes.
	condor.depaul.edu /~scatoiu/seminar/november2001/abstracts.html (1170 words)

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